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Llarull's theorem on noncompact manifolds with boundary

Bo Liu, Daoqiang Liu

Abstract

Recently, Zhang \cite{Zh20} and Li-Su-Wang-Zhang \cite{LSWZ24+} generalized Llarull's theorem to the noncompact complete spin manifold. In this paper, we further extend their results to the noncompact manifold with compact boundary.

Llarull's theorem on noncompact manifolds with boundary

Abstract

Recently, Zhang \cite{Zh20} and Li-Su-Wang-Zhang \cite{LSWZ24+} generalized Llarull's theorem to the noncompact complete spin manifold. In this paper, we further extend their results to the noncompact manifold with compact boundary.
Paper Structure (1 section, 2 theorems, 7 equations, 1 figure)

This paper contains 1 section, 2 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction

Key Result

Theorem 1

Let $(M,g_M)$ be a complete noncompact Riemannian spin manifold. Let $\Phi: M\to \mathbf{S}^m$ be a smooth area decreasing map which is locally constant at infinity and of nonzero degree. Suppose that Then

Figures (1)

  • Figure 2.1: An illustration of noncompact (necessarily not complete) manifold with surgery.

Theorems & Definitions (3)

  • Theorem 1: LSWZ24+,Zh20
  • Theorem A
  • Remark 2